Proof that inverse base-height ratios give equal triangles
Triangle a d e is 3 times higher; triangle c d g has base d g 3 times wider
A tall right-angled figure at the upper right, subdivided by lines, illustrates the proof that two triangles are equal in area when the ratio of their bases is the inverse of the ratio of their heights (axes). Leonardo demonstrates it on two rational triangles: a d e is three times higher than c d g, while c d g has a base d g three times wider than the base d e. Reasoning from the theorem that triangles on the same base stand in the ratio of their axes, he shows the triangle d e c to be one third of both d e a and d g c, so the two triangles come out equal.
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The inverse-proportion rule for equal triangles
Two irrational triangles are equal in area when the base of one to the base of the other stands in the same proportion as the axis of one to the axis of the other, but taken inversely. Leonardo tests the rule on two rational triangles where a d e is three times the height of c d g and c d g has a base three times the width of a d e's.
Proof by triangles on a common base d e
Since triangles on one and the same base stand to each other as their axes, the triangles d e a and d e c on the base d e are triple one of the other. The wider base d g holds three triangles like d e c, so d e c is one third of both d e a and d g c, making the two triangles equal.